Multiplicative deformations of spectrale triples associated to left invariant metrics on Lie groups

We study the triple $(G,\pi ,\text{prs})$ where $G$ is a connected and simply connected Lie group, $\pi$ and $\text{prs}$ are, respectively, a multiplicative Poisson tensor and a left invariant Riemannian metric on $G$ such that the necessary conditions, introduced by Hawkins, to the existence of a non commutative deformation (in the direction of $\pi$) of the spectrale triple associated to $\text{prs}$ are satisfied. We show that the geometric problem of the classification of such triple $(G,\pi ,\text{prs})$ is equivalent to an algebraic one. We solve this algebraic problem in low dimensions and we give the list of all $(G,\pi ,\text{prs})$ satisfying Hawkins’s conditions, up to dimension four.